fisher=NDSolve[{D[u[t,x],t]==D[u[t,x],x,x]+u[t,x]*(1-u[t,x]),u[t,-100]==0,u[t,100]==0,u[0,x]==Exp[-100*x^2]},u,{t,0,100},{x,-100,100}]

Plot[{u[5,x],u[10,x],u[20,x],u[40,x]}/. fisher,{x,-100,100},PlotRange->All]

sol=NDSolve[{D[u[t,x],t]==1*D[u[t,x],x,x],u[0,x]==0,u[t,0]==1-Exp[-1000*t],u[t,5]==0},u,{t,0,100},{x,0,5}]

Plot3D[u[t,x]/. sol,{t,0,10},{x,0,5},PlotRange->All]

Plot[{u[1,x], u[2,x], u[5,x], u[100,x]}/. sol,{x,0,5},PlotRange->All]